Abstract We introduce the notion of an introverted Boolean algebra $\mathcal{B}$ closed-and-open subsets a topological group G, show that associated Stone space $(\nu_{\mathcal{B}} \nu_{\mathcal{B}})$ is totally disconnected semigroup compactification G and every takes this form. identify study universal compactification, semitopological G. Our main results are obtained independently Gelfand th...

Let T X be the full transformation semigroup on the setX and let T X,Y {α ∈ T X : Xα ⊆ Y}. Then T X,Y is a sub-semigroup of T X determined by a nonempty subset Y of X. In this paper, we give a necessary and sufficient condition for T X,Y to be regular. In the case that T X,Y is not regular, the largest regular sub-semigroup is obtained and this sub-semigroup is shown to determine the Green’s re...

We investigate C∗-algebras associated with row-finite topological higherrank graphs with no source, which are based on product system C∗-algebras. We prove the Cuntz-Krieger uniqueness theorem, and provide the condition of simplicity and purely infiniteness of our algebras. We give examples of non-discrete topological higher-rank graphs whose C∗-algebras contain Cuntz’s ax + b-semigroup C∗-alge...

We introduce the notion of path extensions of tiling semigroups and investigate their properties. We show that the path extension of a tiling semigroup yields a strongly F *-inverse cover of the tiling semigroup and that it is isomorphic to an HNN * extension of its semilattice of idempo-tents.